The proof of Birman's conjecture on singular braid monoids

Luis Paris (1)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]

Let B_n be the Artin braid group on n strings with standard generators sigma_1, ..., sigma_{n-1}, and let SB_n be the singular braid monoid with generators sigma_1^{+-1}, ..., sigma_{n-1}^{+-1}, tau_1, ..., tau_{n-1}. The desingularization map is the multiplicative homomorphism eta: SB_n --> Z[B_n] defined by eta(sigma_i^{+-1}) =_i^{+-1} and eta(tau_i) = sigma_i - sigma_i^{-1}, for 1 <= i <= n-1. The purpose of the present paper is to prove Birman's conjecture, namely, that the desingularization map eta is injective.

Domaines: Théorie des groupes [math.GR]; Topologie géométrique [math.GT]

Luis Paris. The proof of Birman's conjecture on singular braid monoids. Geometry and Topology, 2004, 8, pp.1281-1300. ⟨hal-00128156⟩